Signal Reconstruction Method and Apparatus

1. A signal reconstruction method, comprising: determining a correlation between a first residual error and multiple columns in a sensing matrix according to a measured value of an original signal and the sensing matrix; determining a first array most correlative to the measured value of the original signal in the sensing matrix according to the correlation between the first residual error and the multiple columns in the sensing matrix; determining a correlation between a k th residual error and the multiple columns in the sensing matrix according to a correlation between a (k−1) th residual error and the multiple columns in the sensing matrix; determining a k th array most correlative to the measured value of the original signal in the sensing matrix according to the correlation between the k th residual error and the multiple columns in the sensing matrix, wherein 2≦k≦K; and recovering the original signal after determining a K th array most correlative to the measured value of the original signal in the sensing matrix, wherein determining the correlation between the k th residual error and the multiple columns in the sensing matrix according to the correlation between the (k−1) th residual error and the multiple columns in the sensing matrix comprises, from the correlation between the (k−1) th residual error and the multiple columns in the sensing matrix, subtracting a product of a (k−1) th base difference and a (k−1) th difference coefficient to obtain the correlation between the k th residual error and the multiple columns in the sensing matrix, wherein the (k−1) th base difference is a vector or a vector set, and the (k−1) th difference coefficient is a scalar or a scalar set or a set of vectors composed of multiple scalars.

2. The method according to claim 1 , wherein before the from the correlation between the (k−1) th residual error and the multiple columns in the sensing matrix, subtracting a product of a (k−1) th base difference and a (k−1) th difference coefficient to obtain the correlation between the k th residual error and the multiple columns in the sensing matrix, the determining a correlation between a k th residual error and the multiple columns in the sensing matrix further comprises: determining a first base difference when k=2; determining the (k−1) th base difference according to a first base difference when k=3; or determining the (k−1) th base difference according to first to (k−2) th base differences when k≧4.

3. The method according to claim 2 , wherein the determining a first base difference comprises determining the first base difference t N−1 by using formula t 1 =λ 1 ·θ (ρ 1 ) ; and the determining the (k−1) th base difference comprises determining the (k−1) th base difference by using formula t k−1 =−λ k−1 T k−2 ũ k−2 +λ k−1 θ (ρ k−1 ) , wherein λ 1 =√{square root over (1/θ ρ 1 ,ρ 1 )}, θ (ρ 1 ) is a ρ 1 th column of Θ, θ ρ 1 ,ρ 1 is an item in a ρ 1 th row and a ρ 1 th column in Θ, Θ is a correlation matrix of the sensing matrix, ρ 1 is a sequence number of a column in the sensing matrix, wherein the column is a column of a first array most correlative to the measured value of the original signal in the sensing matrix, T 1 =[ t 1 ], T k−2 =[ T k−3 t k−2 ], λ k−1 =1/√{square root over (θ ρ k−1 ,ρ k−1 −ũ k−2 H ũ k−2 )}, ũ k−2 =( t (ρ,:) H ) H , θ ρ k−1 , ρ k−1 is an item in a ρ k−1 th row and a ρ k−1 th column in Θ, t (ρ k ,:) H is a (ρ k−1 ) th row in T k−2 , and ρ k−1 is a sequence number of a column in the sensing matrix, wherein the column is a column of a (k−1) th array most correlative to the measured value of the original signal in the sensing matrix.

4. The method according to claim 3 , wherein before the from the correlation between the (k−1) th residual error and the multiple columns in the sensing matrix, subtracting a product of a (k−1) th base difference and a (k−1) th difference coefficient to obtain the correlation between the k th residual error and the multiple columns in the sensing matrix, the determining a correlation between a k th residual error and the multiple columns in the sensing matrix further comprises determining a k th difference coefficient α k by using formula α k =λ k q ρ k K−1 .

5. The method according to claim 3 , wherein the method further comprises: determining F 1 by using formula F 1 =[λ 1 ]; and determining F k by using formula F k = [ F k - 1 u k - 1 0 k - 1 T λ k ] when 2≦k≦K, wherein u k−1 =−λ k F k−1 ũ k−1 , and wherein the recovering the original signal of a received signal after determining a K th array of the base matrix comprises recovering the signal according to the F K .

6. The method according to claim 5 , wherein the recovering the signal according to the F K comprises: determining a weight coefficient column vector z K by using formula z K =F K a K ; determining x according to the weight coefficient column vector to make {circumflex over (x)} τ k =Z k , wherein z k is a k th item of z K , and τ k is a column in an array most correlative to the measured value of the original signal in the sensing matrix; and obtaining the original signal according to the x.

7. The method according to claim 2 , wherein the determining a first base difference comprises determining a first column and a second column of T 2 as the first base difference, wherein the determining the (k−1) th base difference comprises determining a (2k−3) th column and a (2k−2) th column of T 2k−2 as the (k−1) th base difference, and wherein T 2 =[λ 1 θ (ρ 1 ) f 1,2 θ (ρ 1 ) +λ 2 ×flipud(θ* (ρ 1 ) )], T 2k−2 =[ T 2k−4 λ 2k−3 {tilde over (t)} k (f 2k−3,2k−2 {tilde over (t)} k−1 +λ 2k−2 ×flipud({tilde over (t)}* k−1 ))], λ 1 =√{square root over (1/θ ρ 1 ,ρ 1 )}, θ ρ 1 ,ρ 1 represents an item in a ρ 1 th row and a ρ 1 th column in the sensing matrix Θ, θ (ρ 1 ) is the ρ 1 th column in the sensing matrix Θ, f 1,2 =−λ 2 λ 1 u″ 1 , λ 2 =1/√{square root over (θ ρ 1 ,ρ 1 −(u″ 1 )*u″ 1 )}, u″ 1 =λ 1 ×θ ρ 1 ,N−ρ 1 +1 , θ ρ 1 ,N−ρ 1 +1 represents an item in the ρ 1 th row and a (N−ρ 1 +1) th column in the sensing matrix Θ, q ρ 1,: H is the ρ 1 th row in Q, Q is a matrix composed of correlations between the first residual error and the multiple columns in the sensing matrix, λ 2k−3 =1/√{square root over (θ ρ k−1 ,ρ k−1 −ũ′ 2k−4 H ũ′ 2k−4 )}, ũ′ 2k−4 =( T 2k−4 ( ρ k−1 ,:)) H , T 2k−4 ( ρ k−1 ,:) represents a ( ρ k−1 ) th row in the matrix T 2k−4 , f 2k−3,2k−2 =−λ 2k−2 λ 2k−3 ũ″ 2k−3 , λ 2k−2 =1/√{square root over (1/λ 2k−3 2 −|ũ″ 2k−3 | 2 )}, ũ″ 2k−3 =λ 2k−3 (θ ρ k−1 ,ρ k−1 − T 2k−4 ( ρ k−1 ,:)×ũ′ 2k−4 )*, {tilde over (t)} k−1 =θ (ρ k−1 ) − T 2k−4 ũ′ 2k−4 , and θ (ρ k−1 ) is a ρ k−1 th column in the matrix Θ.

8. The method according to claim 7 , wherein before the from the correlation between the (k−1) th residual error and the multiple columns in the sensing matrix, subtracting a product of a (k−1) th base difference and a (k−1) th difference coefficient to obtain the correlation between the k th residual error and the multiple columns in the sensing matrix, the determining a correlation between a k th residual error and the multiple columns in the sensing matrix further comprises: determining a first row and a second row of A 2 as a first difference coefficient; and determining a (2k−3) th row and a (2k−2) th row of A 2k−2 as the (k−1) th difference coefficient, wherein: A 2 ⁢ k - 2 = [ A ( 2 ⁢ k - 4 ) λ 2 ⁢ k - 3 ⁢ a ~ 2 ⁢ k - 4 , : H ( f 2 ⁢ k - 3 , 2 ⁢ k - 2 ) * ⁢ a ~ 2 ⁢ k - 4 , : H + λ 2 ⁢ k - 2 ⁡ ( a ~ 2 ⁢ k - 4 , : H ) * ] , ⁢ and A 2 = [ λ 1 ⁢ q ρ 1 , : H f 1 , 2 * ⁢ q ρ 1 , : H + λ 2 ⁡ ( q ρ 1 , : H ) * ] .

9. The method according to claim 7 , wherein the method further comprises: determining F 2 by using formula F 2 = [ λ 1 f 1 , 2 0 λ 2 ] ; and determining F 2k by using formula F 2 ⁢ k = [ F 2 ⁢ k - 2 - λ 2 ⁢ k - 1 ⁢ w 2 ⁢ k - 2 ( - f 2 ⁢ k - 1 , 2 ⁢ k ⁢ w 2 ⁢ k - 2 - λ 2 ⁢ k × reorder ⁡ ( w 2 ⁢ k - 2 ) ) 0 2 ⁢ k - 2 H λ 2 ⁢ k - 1 f 2 ⁢ k - 1 , 2 ⁢ k 0 2 ⁢ k - 2 H 0 λ 2 ⁢ k ] when 2≦k≦K, wherein reorder(w 2k−2 ) refers to exchanging positions of a (2i−1) th item and a 2i th item of a vector w 2k−2 , and then calculating a conjugate of each item of the vector, wherein i=1,2, . . . , k−1, and w 2k−2 =F 2k−2 ũ′ 2k−2 , and wherein the recovering the original signal of a received signal after determining a K th column of the base matrix comprises recovering the signal according to the F 2K .

10. A signal reconstruction apparatus, comprising: a memory; and a processor, wherein the memory stores program codes, and wherein the processor is configured to invoke the program codes to perform the following operations: determining a correlation between a first residual error and multiple columns in a sensing matrix according to a measured value of an original signal and the sensing matrix; determining a first array most correlative to the measured value of the original signal in the sensing matrix according to the correlation between the first residual error and the multi s le columns in the sensing matrix; determining a correlation between a k th residual error and the multiple columns in the sensing matrix according to a correlation between a (k−1) th residual error and the multiple columns in the sensing matrix; determining a k th array most correlative to the measured value of the original signal in the sensing matrix according to the correlation between the k th residual error and the multiple columns in the sensing matrix, wherein 2≦k≦K; recovering the original signal after determining a K th array most correlative to the measured value of the original signal in the sensing matrix; and subtracting, from the correlation between the (k−1) th residual error and the multiple columns in the sensing matrix, a product of a (k−1) th base difference and a (k−1) th difference coefficient to obtain the correlation between the k th residual error and the multiple columns in the sensing matrix, wherein the (k−1) th base difference is a vector or a vector set, and the (k−1) th difference coefficient is a scalar or a scalar set or a set of vectors composed of multiple scalars.

11. The apparatus according to claim 10 , wherein the processor is configured to invoke the program codes to perform the following operations: determining a first base difference when k=2; determining the (k−1) th base difference according to a first base difference when k=3; or determining the (k−1) th base difference according to first to (k−2) th base differences when k≧4.

12. The apparatus according to claim 11 , wherein the processor is configured to invoke the program codes to perform the following operations: determining the first base difference t N−1 by using formula t 1 =λ 1 ·θ (ρ 1 ) ; and determining the (k−1) th base difference by using formula t k−1 =−λ k−1 T k−2 ũ k−2 +λ k−1 θ (ρ k−1 ) , wherein λ 1 =√{square root over (1/θ ρ 1 ,ρ 1 )}, θ (ρ 1 ) is a ρ 1 th column of Θ, θ ρ 1 ,ρ 1 is an item in a ρ 1 th row and a ρ 1 th column in Θ, Θis a correlation matrix of the sensing matrix, ρ 1 is a sequence number of a column in the sensing matrix, wherein the column is a column of a first array most correlative to the measured value of the original signal in the sensing matrix, T 1 =[ t 1 ], T k−2 =[ T k−3 t k−2 ], λ k−1 =1/√{square root over (θ ρ k−1 ,ρ k−1 −ũ k−2 H ũ k−2 )}, ũ k−2 =( t ρ,:) H ) H , θ ρ k−1 ,ρ k−1 is an item in a (ρ k−1 ) th row and a (ρ k−1 ) th column in Θ, t (ρ k ,:) H is a (ρ k−1 ) th row in T k−2 , and ρ k−1 is a sequence number of a column in the sensing matrix, wherein the column is a column of a (k−1) th array most correlative to the measured value of the original signal in the sensing matrix.

13. The apparatus according to claim 12 , wherein the processor is configured to invoke the program codes to perform the following operations: determining the k th difference coefficient α k by using formula α k =λ k q ρ k k−1 .

14. The apparatus according to claim 12 , wherein the processor is configured to invoke the program codes to perform the following operations: determining F 1 by using formula F 1 =[λ 1 ]; determining F k by using formula F k = [ F k - 1 u k - 1 0 k - 1 T λ k ] when 2≦k≦K, wherein u k−1 =−λ k F k−1 ũ k−1 ; and recovering the signal according to the F K .

15. The apparatus according to claim 14 , wherein the processor is configured to invoke the program codes to perform the following operations: determining a weight coefficient column vector z K by using formula z K =F K a K ; determining x according to the weight coefficient column vector to make {circumflex over (x)} τ k =z k , wherein z k is a k th item of z K , and τ k is a column in an array most correlative to the measured value of the original signal in the sensing matrix; and obtaining the original signal according to the x.

16. The apparatus according to claim 11 , wherein the processor is configured to invoke the program codes to perform the following operations: determining a first column and a second column of T 2 as a first base difference; and determining a (2k−3) th column and a (2k−2) th column of T 2k−2 as the (k−1) th base difference, wherein T 2 =[λ 1 θ (ρ 1 ) f 1,2 θ (ρ 1 ) +λ 2 ×flipud(θ* (ρ1 ) )], T 2k−2 =[ T 2k−4 λ 2k−3 {tilde over (t)} k (f 2k−3,2k−2 {tilde over (t)} k−1 +λ 2k−2 ×flipud({tilde over (t)}* k−1 ))], λ 1 =√{square root over (1/θ ρ 1 ,ρ 1 )}, θ ρ 1 ,ρ 1 represents an item in a ρ 1 th row and a ρ 1 th column in the sensing matrix Θ, θ (ρ 1 ) is the ρ 1 th column in the sensing matrix Θ, f 1,2 =−λ 2 λ 1 u″ 1 , λ 2 =1/√{square root over (θ ρ 1 ,92 1 −(u″ 1 )*u″ 1 )}, u″ 1 =λ 1 ×θ ρ 1 ,N−ρ 1 +1 , θ ρ 1 N−ρ 1 +1 represents an item in the ρ 1 th row and a (N−ρ 1 +1) th column in the sensing matrix Θ, q ρ 1 ,: H is the ρ 1 th row in Q, Q is a matrix composed of correlations between the first residual error and the multiple columns in the sensing matrix, λ 2k−3 =1/√{square root over (θ ρ k−1 ,ρ k−1 −ũ′ 2k−4 H ũ′ 2k−4 )}, ũ′ 2k−4 =( T 2k−4 ( ρ k−1 ,:)) H , T 2k−4 ( ρ k−1 ,:) represents a ( ρ k−1 ) th row in the matrix T 2k−4 , f 2k−3,2k−2 =−λ 2k−2 λ 2k−3 ũ″ 2k−3 , λ 2k−2 =1/√{square root over (1/λ 2k−3 2 −|ũ″ 2k−3 | 2 )}, ũ″ 2k−3 =λ 2k−3 (θ ρ k−1 ,ρ k−1 − T 2k−4 ( ρ k−1 ,:)×ũ′ 2k−4 )*, {tilde over (t)} k−1 =θ (ρ k−1 ) − T 2k−4 ũ′ 2k−4 , and θ (ρ k−1 ) is a (ρ k−1 ) th column in the matrix Θ.

17. The apparatus according to claim 16 , wherein the processor is configured to invoke the program codes to perform the following operations: determining a first row and a second row of A 2 as a first difference coefficient; and determining a (2k−3) th row and a (2k−2) th row of A 2k−2 as the (k−1) th difference coefficient, wherein: A 2 ⁢ k - 2 = [ A ( 2 ⁢ k - 4 ) λ 2 ⁢ k - 3 ⁢ a ~ 2 ⁢ k - 4 , : H ( f 2 ⁢ k - 3 , 2 ⁢ k - 2 ) * ⁢ a ~ 2 ⁢ k - 4 , : H + λ 2 ⁢ k - 2 ⁡ ( a ~ 2 ⁢ k - 4 , : H ) * ] , ⁢ and A 2 = [ λ 1 ⁢ q ρ 1 , : H f 1 , 2 * ⁢ q ρ 1 , : H + λ 2 ⁡ ( q ρ 1 , : H ) * ] .

18. The apparatus according to claim 16 , wherein the processor is configured to invoke the program codes to perform the following operations: determining F 2 by using formula F 2 = [ λ 1 f 1 , 2 0 λ 2 ] ; determining F 2k by using formula F 2 ⁢ k = [ F 2 ⁢ k - 2 - λ 2 ⁢ k - 1 ⁢ w 2 ⁢ k - 2 ( - f 2 ⁢ k - 1 , 2 ⁢ k ⁢ w 2 ⁢ k - 2 - λ 2 ⁢ k × reorder ⁡ ( w 2 ⁢ k - 2 ) ) 0 2 ⁢ k - 2 H λ 2 ⁢ k - 1 f 2 ⁢ k - 1 , 2 ⁢ k 0 2 ⁢ k - 2 H 0 λ 2 ⁢ k ] when 2≦k≦K, wherein reorder(w 2k−2 ) refers to exchanging positions of a (2i−1) th item and a 2i th item of a vector w 2k−2 , and then calculating a conjugate of each item of the vector, wherein i=1, 2, . . . , k−1, and W 2k−2 =F 2k−2 ũ′ 2k−2 ; and recovering the signal according to the F 2K .